Apparatus and method for bi-directionally sweeping an image beam in the vertical dimension and related apparati and methods

ABSTRACT

A scan assembly of an image generator sweeps an image beam in a first dimension at a first rate and bi-directionally in a second dimension at a slower rate. Sweeping the beam bi-directionally in the vertical dimension (generally the dimension of the lower sweep rate) can reduce the scanning power by eliminating the flyback period, and, where the scan assembly includes a mechanical reflector, can reduce the error in the beam position without a feedback loop by reducing the number of harmonics in the vertical sweep function. Furthermore, because the image beam is “on” longer due to the elimination of the flyback period, the scanned image is often brighter for a given beam intensity. The scan assembly may also sweep the image beam non-linearly in the vertical dimension, and this sweep may be bi-directional or uni-directional. Sweeping the beam non-linearly can also reduce the error in the beam position by reducing the number of harmonics in the vertical sweep function.

CLAIM OF PRIORITY

This application claims priority to U.S. Provisional Application Ser.No. 60/381,569, filed on May 17, 2002, which is incorporated byreference.

BACKGROUND

An electronic image generator, such as television set, scans a viewableimage, or a sequence of viewable video images, onto a display screen byelectronically sweeping an electromagnetic image beam across the screen.For example, in a television set, the image beam is a beam of electrons,and a coil generates a linearly increasing magnetic field or electricfield to sweep the beam.

An optical image generator is similar except that it scans a viewableimage onto a display screen by mechanically sweeping an electromagneticimage beam across the screen. Or, in the case of a Virtual RetinalDisplay (VRD), the optical image generator scans a viewable directlyonto a viewer's retina(s).

FIG. 1 is a view of a conventional optical image-display system 10,which includes an optical image generator 12 and a display screen 14.The image generator 12 includes a beam generator 16 for generating anoptical beam 18, and includes a scan assembly 20 for scanning an imageonto the screen 14 with the beam. Where the system 10 is a VRD, the scanassembly 20 scans the image directly onto a viewer's retina(s) (notshown). The scan assembly 20 includes a reflector 22, whichsimultaneously rotates back and forth in the horizontal (X) and vertical(Y) dimensions about pivot arms 24 a and 24 b and pivot arms 26 a and 26b; respectively. By rotating back and forth, the reflector 22 sweeps thebeam 18 in a two-dimensional (X-Y) raster pattern to generate the imageon the screen 14 (or retina(s)). The scan assembly 20 includes othercomponents and circuitry (not shown) for rotating the reflector 22 andmonitoring its instantaneous rotational position, which is proportionalto the instantaneous location at which the beam 18 strikes the screen14. In an alternative implementation that is not shown, the scanassembly 20 may include two reflectors, one for sweeping the beam 18 inthe horizontal (X) dimension and the other for sweeping the beam in thevertical (Y) dimension. An optical image-display system that is similarto the system 10 is disclosed in U.S. Pat. No. 6,140,979 of Gerhard, etal., entitled SCANNED DISPLAY WITH PINCH, TIMING, AND DISTORTIONCORRECTION and U.S. Pat. No. 5,467,104 of Furness, et al., entitledVIRTUAL RETINAL DISPLAY, each of which is incorporated by reference.

Referring to FIGS. 1-3, the operation of the optical image-displaysystem 10 is discussed.

Referring to FIG. 1, the image generator 12 starts scanning an image atan initial pixel location X=0, Y=0 and stops scanning the image at anend pixel location X=n, Y=m, where n is the number of pixels in thehorizontal (X) dimension of the image and m is the number of pixels inthe vertical (Y) dimension of the image. Specifically, the beamgenerator 16 modulates the intensity of the image beam 18 to form afirst pixel Z_(0,0) of the scanned image when the reflector 22 directsthe beam onto the location X=0, Y=0. As the reflector 22 sweeps the beam18 toward the location X=n, Y=m, the generator 16 periodically modulatesthe intensity of the beam to sequentially form the remaining pixels ofthe image including the last pixel Z_(n,m). Then, the image generator 12starts scanning the next image at the location X=0, Y=0, and repeatsthis procedure for all subsequent images.

Referring to FIG. 2, during the scanning of the image, the reflector 22sinusoidally sweeps the beam 18 bi-directionally in the horizontal (X)dimension at a horizontal sweep frequency f_(h)=1/t_(h), where t_(h) isthe period of the horizontal sinusoid. FIG. 2 is a plot of thishorizontal sinusoid, which indicates the position of the beam 18 in thehorizontal (X) dimension versus time, where + corresponds to the rightside of the screen 14 and − corresponds to the left side. As this plotshows, the reflector 22 oscillates in a sinusoidal manner about thepivot arms 24 a and 24 b at f_(h), and thus sinusoidally sweeps the beam18 from side to side of the screen 14 at the same frequency. Thehorizontal sweep is bi-directional because the beam 18 is “on”, and thusgenerates pixels, in both the left-to-right (+X) and right-to-left (−X)horizontal directions. Although not required, f_(h) may substantiallyequal the resonant frequency of the reflector 22 about the arms 24 a and24 b. One advantage of designing the reflector 22 such that it resonatesat f_(h) is that the scan assembly 20 can drive the reflector in thehorizontal (X) dimension with relatively little power.

Referring to FIG. 3, the reflector 22 also linearly sweeps the beam 18uni-directionally in the vertical (Y) dimension at a vertical sweepfrequency f_(v)=1/t_(v), where t_(v) is the period of the verticalsaw-tooth wave. FIG. 3 is a plot of this saw-tooth wave, which indicatesthe position of the beam 18 in the vertical (Y) dimension versus time,where + corresponds to the bottom of the screen 14 and − corresponds tothe top. As this plot shows, during a vertical scan period V, the scanassembly 20 linearly rotates the reflector 22 about the pivot arms 26 aand 26 b from a top position to a bottom position, thus causing thereflector to sweep the beam 18 from the top pixel Z_(0,0) of the screen14 to the bottom (pixel Z_(n,m)) of the screen (−Y direction). During afly-back period FB, the scan assembly 20 quickly (as compared to thescan period V) rotates the reflector 22 back to its top position(Z_(0,0)) to begin the scanning of a new image. Consequently, t_(v)=V+FBsuch that the vertical sweep frequency f_(v)=1/(V+FB). Moreover, thevertical sweep is unidirectional because the beam 18 is “on” only duringthe scan period V while the reflector 22 sweeps the beam from top(Z_(0,0)) to bottom (Z_(n,m)) (−Y direction), and is off during theflyback period FB when the reflector 22 returns to its top position(Z_(0,0)). One advantage of vertically sweeping the beam linearly anduni-directionally is that this is compatible with conventional videoequipment that generates video images for display using this samevertical sweeping technique.

Unfortunately, uni-directionally sweeping the beam 18 in the vertical(Y) dimension may increase the cost, complexity, size, and powerconsumption of the system 10. Referring to FIG. 3, the vertical-sweepsaw-tooth wave includes many harmonics of the fundamental vertical sweepfrequency f_(v). For example, if f_(v)=60 Hz, then the saw-tooth wavehas significant harmonics up to approximately 3600 Hz (the 60^(th)harmonic, i.e., 60×f_(v)). The vibrations that these higher harmonicsintroduce to the reflector 22 may cause a significant error in thevertical (Y) location of the beam 18. That is, the reflector 22 may notrotate smoothly through the vertical scan, producing a vertical “jitter”or “ripple” that may cause the location where the beam 18 strikes thescreen 14 to be misaligned with the location of the pixel Z that thebeam is currently forming. One way to reduce or eliminate this error isto include a feedback loop (not shown in FIG. 1) in the scan assembly 20to smoothen the rotation of the reflector 22 during the vertical scanperiod V. Such a feedback loop is disclosed in U.S. Pat. No. 6,140,979,which is incorporated by reference. Unfortunately, such a feedback loopoften includes complex circuitry that can occupy significant layoutarea, and, thus, may increase the complexity, size, and cost of theimage generator 12. Furthermore, quickly rotating the reflector 22 fromits bottom position (Z_(n,m)) to its top position (Z_(0,0)) during theflyback period FB often requires that the scan assembly 20 drive theelectromagnets (not shown) that rotate the reflector 22 with asignificant peak current. Unfortunately, this may increase the powerconsumed by the image generator 12 and the size of the scan assembly'scurrent-driver circuit (not shown), and thus may further increase thecost of the image generator.

Another way to reduce or eliminate the ripple error is to generate adrive signal that offsets the non-linearity of the vertical scan. Avariety of approaches can be applied to reduce the ripple.

In one such approach, a feedback loop in the scan assembly 20 comparesthe detected angular position about the vertical axis with an idealizedwaveform. The loop then generates a drive signal to minimize the error,according to conventional feedback control approaches and smoothen therotation of the reflector 22 during the vertical scan period V.

In another approach, a general analytical or empirical model of thevertical scan assembly is developed for the general characteristics ofthe scan assembly 20, using parameters of the a set of scan assemblies.Then, for the specific scan assembly 20 in use, the individual responseis characterized during manufacture or at system start-up to refine themodel parameters more precisely and the data representing the particularscan assembly 20 are stored in memory. The scan assembly then generatesa drive signal according to the stored model to minimize the ripple.

In some cases, such feedback loops and adaptive control systems mayinclude complex circuitry that occupies significant layout area orrequire specialized components, and thus may increase the complexity,size, and cost of the image generator 12.

SUMMARY

According to an embodiment of the invention, a scan assembly sweeps animage beam in a first dimension at a first frequency andbi-directionally in a second dimension at a second frequency that isless than the first frequency.

For example, sweeping the beam bi-directionally in the verticaldimension can reduce the scanning power by eliminating the flybackperiod, and can reduce error in the beam position without a feedbackloop by reducing the number of harmonics in the vertical sweep function.Furthermore, because the image beam is “on” longer due to theelimination of the flyback period, the scanned image is often brighterfor a given beam intensity, thus allowing one to proportionally reducethe intensity, and thus the power, of the image beam for a given imagebrightness.

According to another embodiment of the invention, a scan assembly sweepsan image beam in the first dimension and non-linearly in the seconddimension.

For example, sweeping the beam non-linearly in the vertical dimensioncan also reduce the error in the beam position by reducing the number ofharmonics in the vertical sweep function.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a conventional optical image-display system.

FIG. 2 is a plot of a sinusoid that indicates the position of the imagebeam of FIG. 1 in the horizontal dimension versus time.

FIG. 3 is a plot of a saw-tooth wave that indicates the position of theimage beam of FIG. 1 in the vertical dimension versus time.

FIG. 4 is a plot of a bi-sinusoidal image-scanning pattern overlaid on asource-image grid pattern according to an embodiment of the invention.

FIG. 5A is a plot of the horizontal and vertical sweeping sinusoidsversus time, where the sinusoids have an preferred phase relationshipthat yields the scanning pattern of FIG. 4 according to an embodiment ofthe invention.

FIG. 5B is a plot of the horizontal and vertical sweeping sinusoidsversus time, where the sinusoids have another preferred phaserelationship that also yields the scanning pattern of FIG. 4 accordingto an embodiment of the invention.

FIG. 6 is a plot of the bi-sinusoidal scanning pattern of FIG. 4, wherethe phase relationship between the horizontal and vertical sweepingsinusoids is not optimal according to an embodiment of the invention.

FIG. 7 is a plot of the bi-sinusoidal scanning pattern of FIG. 4, wherethe phase relationship between the horizontal and vertical sweepingfunctions is worst case according to an embodiment of the invention.

FIG. 8 is a plot of the horizontal and vertical sweeping functionsversus time, where the sinusoids have the worst-case phase relationshipthat yields the scanning pattern of FIG. 7 according to an embodiment ofthe invention.

FIG. 9 is a sequence of video images that are bi-sinusoidally scanned ina manner that causes a viewer to perceive false ghost objects.

FIG. 10 is a sequence of video images that are bi-sinusoidally scannedin a manner that reduces or eliminates a viewer's perception of falseghost objects according to an embodiment of the invention.

FIG. 11 is a plot of a bi-sinusoidal scanning pattern that may cause theresulting scanned image to have a non-uniform brightness.

FIG. 12 is a plot of horizontal and vertical sweeping functions versustime, where the vertical sweeping function is modified according to anembodiment of the invention to improve the brightness uniformity of aresulting scanned image.

FIG. 13 is a plot of the scanning and grid patterns of FIG. 4 andillustrates a technique for interpolating the pixels of a scanned imagefrom the pixels of a corresponding source image according to anembodiment of the invention.

FIG. 14 is a portion of the scanning and grid patterns of FIG. 13 andillustrates a technique for interpolating the pixels of a scanned imagefrom the pixels of a corresponding source image according to anotherembodiment of the invention.

FIG. 15 is a block diagram of an interpolation circuit that caninterpolate the pixels of a scanned image using the techniquesillustrated in FIGS. 13 and 14 according to an embodiment of theinvention.

FIG. 16 is a plot of the horizontal and vertical sweeping sinusoids ofFIG. 5A and line segments that the interpolation circuit of FIG. 15 usesto linearly approximate the sweeping sinusoids according to anembodiment of the invention.

FIG. 17 is a block diagram of an image generator that can function asdescribed above in conjunction with FIGS. 4-16 according to anembodiment of the invention.

DETAILED DESCRIPTION Bi-Sinusoidal Scanning Pattern

Referring to FIGS. 4-8, one general embodiment according to theinvention is a scan assembly, similar to the scan assembly 20 (FIG. 11),that bi-directionally sweeps the image beam in the vertical (Y)dimension. That is, the image beam is “on” while the scan assemblysweeps the beam from the top to the bottom of the screen, and is also“on” while the scan assembly sweeps the beam from the bottom back to thetop of the screen.

Still referring to FIGS. 4-8, in one embodiment the scan assembly sweepsan image beam sinusoidally and bi-directionally in both the horizontal(X) and vertical (Y) dimensions, although one can use a sweepingfunction other than a sinusoid in either of the horizontal and verticaldimensions as discussed below in conjunction with FIG. 12. For clarity,“bi-sinusoidal” is used to denote a sinusoidal and bi-directional sweepof the image beam in both the horizontal (X) and vertical (Y)dimensions. Because both of the horizontal and vertical sweep functionsare sinusoids, the resulting two-dimensional scan pattern is a repeatedpattern such as a Lissajous pattern. For simplicity of presentation, theterm Lissajous pattern will be used herein to refer to patterns thatemploy sinusoidal motion in about two or more axes.

The following variables represent the parameters used to define abi-sinusoidal sweep of an image beam according to this embodiment of theinvention.

X(t)=the horizontal sweep sinusoid as a function of time

Y(t)=the vertical sweep sinusoid as a function of time

f_(h)=horizontal sweep frequency

f_(v)=vertical sweep frequency

φ_(h)=initial phase of the horizontal sweep sinusoid (X(t)

φ_(v)=initial phase of the vertical-sweep sinusoid (Y(t)

A=frequency at which the Lissajous scan pattern formed by the horizontaland vertical sweep sinusoids repeats itself

R=frequency at which images are to be scanned/displayed

N=number of images to be scanned/displayed during a period 1/A

n_(h)=number of cycles of the horizontal sweep sinusoid per period 1/A

n_(v)=number of cycles of the vertical sweep sinusoid per period 1/A

p_(h)=horizontal resolution of, i.e., number of horizontal pixels in,the source and scanned images

p_(v)=vertical resolution of, i.e., number of vertical pixels in, thesource and scanned images

Δ=maximum width between scan lines in the dimension of the lowest sweepfrequency.

And these parameters are defined or related by the following equations.As discussed below, some of these equations are not absolute, but aremerely guidelines.X(t)=(p _(h)/2)sin(2πf _(h) t+φ _(h))  (1)Y(t)=(p _(v)/2)sin(2πf _(v) t+φ _(v))  (2)For example, if p_(h)=800 and p_(v)=600, X(t) ranges from +400 pixels(400 pixels from the center to the right side of the screen 14 inFIG. 1) to −400 pixels (400 pixels from the center to the left side ofthe screen), and Y(t) ranges from +300 pixels (300 pixels from thecenter to the top of the screen) to −300 pixels (300 pixels from thecenter to the bottom of the screen).N=R/A  (3)For example, if the Lissajous pattern repeats itself at a rate of A=1Hz, and the images are to be scanned out at a rate of R=5 Hz, thenN=5/1=5 images are displayed per each period 1/A during which a completeLissajous pattern is scanned.f_(h)=An_(h)  (4)f_(v)=An_(v)  (5)For example, if A=1 Hz and it takes n_(h)=9 periods of the horizontalsweep frequency f_(h) to complete the Lissajous pattern, thenf_(h)=1×9=9 Hz. Similarly, if n_(v)=2 periods of the vertical sweepfrequency f_(v) to complete the Lissajous pattern, then f_(v)=1×2=2 Hz.It is preferred, but not required, that n_(h) and n_(v) be integers thathave no common factor between them other than one. As discussed below inconjunction with FIG. 4, if n_(h) and n_(v) do have a common factorother than one, then f_(h) and f_(v) are higher than they have to be fora given maximum line width Δ.

Combining equations (4) and (5) yields:f _(h) /n _(h) =f _(v) /n _(v) =A  (6)

Furthermore, assuming that f_(v) is less than f_(h), which is typically(but not always) the case, then:Δ=(πp _(v) A)/2f _(h)  (7)

As discussed below in conjunction with FIGS. 4-8, because most sourceimages assume pixels that are arranged in a grid pattern, a designer ofa bi-sinusoidal image generator would be led to select values for theabove parameters so that the resulting Lissajous pattern of the scannedimage “fits” the grid pattern. For example, source images that arecomputer generated or that are captured by a conventional video cameraor digital camera typically have their pixels arranged in a gridpattern. Although the Lissajous scan pattern may differ significantlyfrom the grid pattern, the quality of the scanned image can approach orequal that of the source image by proper selection of values for theabove parameters. Of course if the pixels of the source image arearranged in a Lissajous pattern, then the designer can merely select theparameter values such that the Lissajous scan pattern of the scannedimage is the same as the Lissajous pattern of the source image.

FIG. 4 is a plot of an example of one bi-sinusoidal scanning pattern 40overlaid on a source-image grid pattern 42 according to an embodiment ofthe invention. In this example, p_(h)=8, p_(v)=6, n_(h)=9, and n_(v)=2.The bi-sinusoidal scanning pattern 40 represents the path of the imagebeam as it sweeps horizontally and vertically to scan an image, and thusrepresents all possible locations of the pixels that compose the scannedimage. Conversely, the intersecting points of the grid pattern 42identify the locations of the pixels P_(n,m) that compose the sourceimage, where n=P_(h) and m=P_(v). By convention, where p_(h) and p_(v)are even numbers as in this example, the centers C of the scanned andoriginal images are coincident, and are respectively located ±0.5 pixelsfrom P_(n,−1) and P_(n,1) in the vertical (Y) dimension and ±0.5 pixelsfrom P_(−1,m) and P_(1,m) in the horizontal (X) dimension. Consequently,in this example, the respective distances ±D_(v) from the center C tothe tops 44 and the bottoms 46 of the source and scanned images equalP_(v)/2=M/2=±3 pixels, and the respective distances ±Dh to the left 48and right 50 of the source and scanned images equal P_(n)/2=n/2=±4pixels. This is consistent m/2 with equations (1) and (2), where thepeak amplitude of the vertical sinusoid Y(t) equals p_(v)/2=m/2 6/2=3pixels, and the peak amplitude of the horizontal sinusoid X(t) equalsp_(h)/2=h/2 8/2=4 pixels. Also in this example, it is assumed that animage generator (FIG. 17) includes a scan assembly for driving areflector, or other beam deflector, bi-sinusoidally in the vertical (Y)dimension as discussed below.

Still referring to FIG. 4, to design an image generator for scanning thepattern 40, a designer first determines the desired maximum line widthΔ. As discussed above and as shown in FIG. 4, Δ is the maximum width inthe vertical (Y) dimension between two adjacent horizontal lines of thescan pattern 40. Empirical studies of image quality indicate that Δ≦˜1can be a desirable selection. Therefore, setting Δ≦1 to satisfy thisguideline, the following expression for f_(h) is derived from equation(7) above:f _(h)≧(πp _(v) A)/2  (7)

Next, the A can be determined. For example, assume that the sourceimages are video images having a display rate of R=30 Hz (30 images persecond), and that image generator (FIG. 17) is to scan one image perLissajous pattern (N=1). Therefore, A=30 Hz per equation (3).

Then, the designer selects n_(v). For example, assume that the designerwishes n_(v)=2 (two vertical sweep cycles per each Lissajous pattern).

Next, the designer calculates f_(v) from equation (5). In this example,A=30 Hz and n_(v)=2, giving f_(v)=60 Hz. Although f_(v) can be anyfrequency compatible with the scan assembly, it has been empiricallydetermined that, for image quality purposes, ˜50 Hz≦f_(v)≦˜75 Hz orf_(v)>1500 Hz.

Then, the designer calculates the minimum value of f_(h) from equation(8). In this example where P_(v)=6 and A=30 Hz, f_(h)≧(π6×30)/2≧˜282.60Hz.

Next, the designer preferably chooses the lowest value of f_(h) thatsubstantially satisfies equation (8) and that yields an integer forn_(h) that has no common factor with n_(v) other than one. From equation(6), selecting f_(h)=270 Hz yields n_(h)=9, which has no common integerfactor with n_(v)=2. Although f_(h)=270 Hz<282.60 Hz, it yields, perequation (7), a maximum line width Δ=1.05 pixels, which is within 5% ofthe desired maximum line width of 1 pixel. Therefore, f_(h)=270 Hzsubstantially satisfies equation (8). Of course, the designer can selecta lower value for f_(h) if it yields a scanned image having anacceptable quality. Alternatively, the designer may select a highervalue for f_(h), such as 330 Hz, which yields n_(h)=11 and Δ<1. But thescan assembly (FIG. 17) typically consumes less power at a lowerhorizontal frequency f_(h).

Other embodiments of the above-described design technique arecontemplated. For example, the designer may perform the steps of thedesign procedure in an order that is different than that describedabove. Furthermore, n_(h) and n_(v) may have a common factor other thanone. However, this merely results in higher frequencies f_(h) and f_(v)with no decrease in the maximum line width Δ. For example, the designercould select f_(h)=540 Hz and f_(v)=120 Hz such that n_(h)=18 andn_(v)=4. But these higher frequencies would merely retrace the Lissajouspattern 40 twice as fast as f_(h)=270 Hz and f_(v)=60 Hz. Therefore, asdiscussed above, selecting n_(h) and n_(v) to have no common non-unityfactor typically provides the smallest Δ for the frequencies used.Moreover, one or both of n_(h) and n_(v) may be a non-integer. This,however, may cause the Lissajous pattern to begin and end at differentpoints on the display screen from retrace period (1/A) to retraceperiod, thus causing the pattern to “roll”, unless addition processingis applied. Such rolling may adversely affect the quality of the scannedimage. In addition, although in the above example f_(h)>>f_(v), thedesigner can select f_(v)≈f_(h), f_(v)>f_(h), or f_(v)>>f_(h). Wheref_(v)>>f_(h), the designer should substitute p_(h) for p_(v) and f_(v)for f_(h) in equation (7), and where f_(v)≈f_(h), the designer shoulduse equation (7) and its vertical equivalent to insure the desiredmaximum line width Δ in both the horizontal (X) and vertical (Y)dimensions. Furthermore, the horizontal and vertical sweep functionsX(t) and Y(t) may be other than sinusoids. An example of anon-sinusoidal function Y(t) is discussed below in conjunction with FIG.12.

While the above-described rolling may degrade image quality or increasecomplexity of the data processing, such an approach may be desirable insome cases. For example, in imaging applications or lower-resolutionapplications, a non-integer ratio can allow greater flexibility inscanner design or can increase addressability, while typicallyincreasing the risk of image artifacts.

Referring to FIGS. 4, 5A, and 5B, the designer next determines apreferred phase relationship between the horizontal sinusoid X(t) andvertical sinusoid Y(t) of equations (1) and (2), that yields thetheoretical minimum for the maximum line width Δ calculated according toequation (7). FIG. 5A is a plot of X(t) and Y(t) versus time for onepossible preferred phase relationship that yields the Lissajous scanpattern 42 of FIG. 4, and FIG. 5B is a plot of X(t) and Y(t) versus timefor another possible preferred phase relationship that yields thepattern 42.

Generally, as discussed below in conjunction with FIGS. 6-8, a preferredphase relationship occurs when there is minimum correlation between thepeaks of X(t) and Y(t). Specifically, a preferred phase relationshipexists between X(t) and Y(t) when both of the following equations aresimultaneously satisfied:2πf _(v) t+φ _(v0)(the total phase of Y(t))=±π/2  (9)2πf _(h) t+φ _(h0)(the total phase of X(t))=−π/2+(π/n _(v))[k=½] fork=0,1, . . . (2n _(v)−1)  (10)As a corollary to equations (9) and (10), a preferred phase relationshipalso exists between X(t) and Y(t) when both of the following equationsare simultaneously satisfied:2πf _(h) t+φ _(h0)(the total phase of X(t))=±π/2  (11)2πf _(h) t+φ _(v0)(the total phase of Y(t))=−π/2+(π/n _(h))[k+½] fork=0,1, . . . (2n _(h)−1)  (12)Because fv≠fh, the instantaneous difference between the total phases ofX(t) and Y(t) changes over time. Therefore, equation (10) definesallowable phases of X(t) when the phase of Y(t) has a given value—±π/2in this example; similarly, equation (12) defines allowable phases ofY(t) when the phase of X(t) has a given value—again ±π/2. One can alsoderive other equations that yield allowable phases for X(t) when Y(t)has a given phase other than ±π/2, or that yield allowable phases forY(t) when X(t) has a given phase other than ±π/2. But regardless ofwhich equations are used, they all define the same preferred phaserelationship(s) shown in FIGS. 5A and 5B.

Still referring to FIGS. 4, 5A, and 5B, to illustrate the concept ofsuch a preferred phase relationship, equation (12) is solved for theX(t) and Y(t) of FIGS. 5A and 5B, where n_(h)=9. Specifically, accordingto equation (12), a preferred phase relationship exists when for eachpeak (phase=±π/2) of X(t), the total phase of Y(t) is given by TABLE I:

TABLE I k Total Phase of Y(t) 4  0π   5  π/9 6  2π/9 7  3π/9 8  4π/9 9 5π/9 10  6π/9 11  7π/9 12  8π/9 13  π   14 10π/9 15 11π/9 16 12π/9 1713π/9 0 14π/9 1 15π/9 2 16π/9 3 17π/9

As shown in FIGS. 5A and 5B, for each peak of X(t), the total phase ofY(t) does indeed equal one of the values in TABLE I. More specifically,FIG. 5A shows that a first preferred phase relationship exists when thetotal phase of Y(t) equals a respective odd multiple of π/9 for eachpeak of X(t), and FIG. 5B shows that a second preferred phaserelationship exists when the total phase of Y(t) equals a respectiveeven multiple of π/9 for each peak of X(t). Although both the first andsecond preferred phase relationships yield the scan pattern 40 (FIG. 4),the scan assembly (FIG. 17) scans the pattern 40 by sweeping the imagebeam in a first direction for the first preferred phase relationship andin the opposite direction for the second preferred phase relationship.But because the sweep direction typically does not affect the quality ofthe scanned image, either preferred phase relationship typically yieldsan acceptable scanned image.

Referring to FIGS. 6-8, the undesirable effects of shifting the phaserelationship between X(t) and Y(t) from a preferred phase relationshiptoward and to a worst case is discussed. Specifically, shifting thephase relationship causes an undesirable increase in the maximum linewidth Δ (FIGS. 6 and 7) from its theoretical minimum value (FIG. 4).

FIG. 6 is a plot of the bi-sinusoidal scanning pattern 40 of FIG. 4,where the phase relationship between X(t) and Y(t) is not preferred, andthus where the maximum line width Δ is larger than its theoreticalminimum value (FIG. 4). The scanning pattern 40 has two components.During a first cycle of the vertical sweep function Y(t), the scanassembly (FIG. 17) sweeps the first component, and during a second cycleof Y(t), the scan assembly sweeps the second component, which isspatially offset from the first component. As the phase relationshipbetween X(t) and Y(t) begins to shift from the preferred, the first andsecond components effectively move toward each other, thus increasingthe maximum line width Δ. While the preferred phase relationship may beconsidered optimal, the system can be operated away from this optimumwhere other systems considerations, such as cost, make such operationdesirable. The increased maximum line width Δ may introduce imageartifacts, though these may be acceptable for some applications.

FIG. 7 is a plot of the bi-sinusoidal scanning pattern 40 of FIG. 4,where the phase relationship between X(t) and Y(t) is worst case, andthus where the maximum line width Δ has its largest value. Theworst-case phase relationship effectively causes the first and secondcomponents of the pattern 40 to merge such that they overlap one andother. During a first cycle of the vertical sweep function Y(t), thescan assembly (FIG. 17) sweeps the first component from the top left ofthe pattern 40 to the top right. And during the second cycle of Y(t),the scan assembly sweeps the second component by retracing the firstcomponent from the top right of the pattern 40 to the top left. That is,the scan assembly effectively sweeps the pattern 40 in one directionduring the first cycle of Y(t) and retraces the pattern in the otherdirection during the second cycle of Y(t). Because the two componentsare overlapping, the resulting worst-case Lissajous pattern 40 isequivalent to a single-component pattern having n_(v)=1, n_(h)=4.5, anda maximum line width Δ≈2 pixels, which is twice the theoretical minimumof ˜1 pixel as shown in FIG. 4.

FIG. 8 is a plot of X(t) and Y(t) versus time for a first worst-casephase relationship that yields the Lissajous pattern 40 of FIG. 7. Aworst-case phase relationship occurs when there is maximum correlationbetween the peaks of X(t) and Y(t). More specifically, a worst-casephase relationship occurs when a peak of X(t) periodically coincideswith a peak of Y(t). For example, at times t1 and t2, the positive peaksof Y(t) respectively coincide with a negative and a positive peak ofX(t) to yield the first worst-case phase relationship. These peakcoincidences respectively correspond to the top left and top right ofthe pattern 40, where the scan assembly effectively “bounces” the imagebeam into and back out of the top left and top right “corners” of thepattern 40.

Referring to FIGS. 7 and 8, a second worst-case phase relationshipoccurs when the negative peaks of Y(t) respectively coincide with anegative and a positive peak of X(t), and thus yields a Lissajous scanpattern that is upside down relative to the pattern 40.

Therefore, referring to FIGS. 4-8, each preferred phase relationship forX(t) and Y(t) is exactly half-way between two respective worst-casephase relationships, and equations (9)-(12) yield these half-way points.Specifically, where n_(h)=9 and n_(v)=2, for the first and secondworst-case phase relationships the total phase of Y(t) is an oddmultiple of π/18 for each peak (±π/2) of X(t) (see FIG. 8), and for thefirst and second preferred phase relationships, the total phase of Y(t)is an even multiple of π/18 for each peak of X(t) per equation (12).Consequently, because the even multiples of π/18 are exactly half-waybetween the odd multiples of π/18, the two preferred phase relationshipsare half-way between the two worst-case phase relationships.

Offsetting a Source-Image Switch Rate f_(s) from the Vertical SweepFrequency f_(v) for a Bi-Directional Vertical Sweep

As discussed below in conjunction with FIGS. 9 and 10, a viewer (notshown) may perceive artifacts such as false ghost objects when videoimages are scanned in a temporal sequence that is different from thetemporal sequence of the corresponding source images. More specifically,the human eye may perceive such artifacts in video images that arebi-directionally scanned in the vertical dimension if the rate fs atwhich the image beam is switched from one source image to the other issynchronized to the vertical sweep frequency f_(v). A viewer perceives atrue ghost object when he views an object that is moving faster than theimage persistence of his eye. Specifically, when the eye perceives anobject, the image of the object persists for a certain period of time,which is approximately a few milliseconds, even after the object movesfrom the location in which the eye initially perceived it. If the objectis moving fast enough, then the eye perceives a “blur”, which isequivalent to simultaneously perceiving the object in multiplelocations. One can observe this phenomenon for by moving a fingerquickly back and forth while attempting to view the finger. “Ghostobject” is merely another name for this blur, and refers to the eye'sperception of an object in one or more locations that the object doesnot occupy. A false ghost object is a ghost object that a viewerperceives in a sequence of video images but that he would not perceiveif he had viewed the object directly. Typically, false ghost objects arecaused by errors introduced while capturing or scanning an image.

FIG. 9 is a view of three sequential scanned video images 50 a-50 c,where the bi-directional scanning of the images in the verticaldimension may cause a viewer (not shown) to perceive false ghost objects52 and 54.

The bi-sinusoidally scanned images 50 a-50 c each correspond to arespective source video image S1-S3, and depict motion of a ball 56 anda toy car 58. That is, the image 50 a is a bi-sinusoidally scannedreproduction of S1, the image 50 b is a bi-sinusoidally scannedreproduction of S2, and the image 50 c is a bi-sinusoidally scannedreproduction of S3. The image generator (FIG. 17) may receive the pixelsof the source images S from an image buffer (FIG. 17) or in real timevia a stream of video data.

The source images S1-S3 are captured such that a known time elapsesbetween the capture of a moving object from one source image to thenext. Specifically, if the source images S are captured by aconventional raster scan or light-integration technique, then theelapsed time between the capture of pixels in the same relative locationof successive source images S is substantially constant. For example, ifthe source images S1-S3 are so captured at a rate of 30 Hz ( 1/30^(th)second per image), then the elapsed time between the pixels P1 in S1 andP2 in S2 equals 1/30 seconds, as does the elapsed time between thepixels P3 in S2 and P4 in S3. Consequently, the relative distancebetween the positions of the ball 56 in the source images S1 and S2represents the movement of the ball during the approximately 1/30^(th)of a second that elapses between the capture of the ball in these twopositions. Similarly, the relative distance between the positions of thecar 58 in the source images S2 and S3 represents the movement of the carduring the approximately 1/30 of a second that elapses between thecapture of the car in two positions.

But by repeatedly switching from one source image S to the next sourceimage S in the same relative location of the images 50, an imagegenerator (FIG. 17) that scans the images 50 bi-directionally in thevertical dimension may generate sequential occurrences of a movingobject so quickly that the eye perceives a false ghost object in thisrelative location. In the example of FIG. 9, f_(s)=2f_(v) because theimage generator switches the image beam from the source image S1 to thesource image S2 at the top of the scanned image 50 b, switches the beamfrom S2 to S3 at the bottom of the image 50 b, and repeats thisswitching pattern for subsequent images S and 50. Consequently, assumingthat f_(v)=15 Hz, and each image 50 is scanned in 1/30^(th) of a second,and the ball 56 is approximately ⅛ of the way down from the tops of theimages 50, the time t that elapses between the generation of the ball 56in the image 50 a and the generation of the ball in the image 50 b isapproximately ¼× 1/30= 1/120^(th) of a second, which is significantlyless than the actual 1/30^(th) of a second between the positions of theball in the source images S1 and S2. Therefore, if T is less than thepersistence of the human eye, then as one views the scanned image 50 b,he perceives the ball 56 as being in its S1 and S2 locationssimultaneously, where his persistent perception of the ball from theimage 50 a gives rise to the false ghost object 52 in the image 50 b.Looking at this another way and using the above example to illustrate,the vertical bi-directional scan effectively increases the perceivedspeed of the ball 56 fourfold by “tricking” the viewer into perceivingthat the ball moved between its positions in 50 a and 50 b in 1/120^(th)second instead of in the actual 1/30^(th) of a second. And even if theball 56 was traveling fast enough to generate a true ghost object, theabove-described phenomenon may still generate a false ghost object 52 byexacerbating the true ghost object. Similarly, if the time that elapsesbetween the generation of the car 58 in the images 50 b and 50 c T isless than the persistence of the human eye, then as the viewer views thescanned image 50 c, he perceives the car 58 as being in its 50 b and 50c locations simultaneously. Consequently, this persistent perception ofthe car from the image 50 b gives rise to the false ghost object 54 inthe image 50 c.

Still referring to FIG. 9, one way to reduce or eliminate the perceptionof false ghost objects is to scan the images 50 a in only one verticaldirection. For example, instead of scanning the image 50 a from bottomto top during a first half of a first vertical sweep cycle and scanningthe image 50 b from top to bottom during the second half of the firstcycle, the scan assembly (FIG. 17) can scan the image 50 a from bottomto top during the first half of the first vertical cycle, inactivate theimage beam during the second half of the first cycle, and then scan theimage 50 b from bottom to top during the first half of a second verticalcycle. As long as f_(v) is not much greater than the frequency at whichthe images S1-S3 are captured, this uni-directional vertical scansubstantially eliminates false ghost objects.

Referring to FIG. 10, another technique for reducing or eliminatingfalse ghost objects is to select f_(s) such that it is out of sync withf_(v), i.e., such that the image generator (FIG. 17) does not frequentlyswitch from one source image S to another at the same relative locationof the scanned images 50.

For example, where f_(s)=8f_(v)/5, the image generator (FIG. 17) firstgenerates the image beam from the pixels of the source image S1 as itscans the image 50 a from the bottom 60 a to the top 62 a.

Then, the image generator starts scanning the image 50 b from the top 62b, but does not begin generating the beam from the pixels of the sourceimage S2 until line 64 b, which is ¼ way down the image 50 b and 5/4images 50 down from the bottom 60 a. That is, the top ¼ of the image 50b is the same as the top ¼ of the image 50 a because the image generatorcontinues generating the image beam from the source image S1 until line64 b of the image 50 b.

Next, the image generator finishes scanning the image 50 b from the line64 b downward while generating the image beam from pixels of the sourceimage S2.

Then, the image generator starts scanning the image 50 c from the bottom60 c, but does not switch the generation of the beam to the pixels ofthe source image S3 until the line 66 c, which is ½ way up the image 50c and 5/4 images 50 from the line 64 b. That is, the bottom ½ of theimage 50 c is the same as the bottom ½ of the image 50 b because theimage generator continues generating the image beam from the sourceimage S2 until the line 66 c of the image 50 c.

Next, the image generator finishes scanning the image 50 c from the line66 c upward while generating the image beam from pixels of the sourceimage S3.

Then, the image generator starts scanning the image 50 d from the top 62d, but does not switch the generation of the beam to the pixels of thesource image S4 until the line 68 d, which is ¾ the way down the image50 c and 5/4 images 50 from the line 66 c. That is, the top ½ of theimage 50 d is the same as the top ½ of the image 50 c because the imagegenerator continues generating the image beam from the source image S3until the line 68 of the image 50 d.

Next, the image generator finishes scanning the image 50 d from the line68 d downward while generating the image beam from pixels of the sourceimage S4.

Then, the image generator starts scanning the image 50 e from the bottom60 e, but does not switch the beam to the pixels of the source image S5until the top 62 e of the image 50 e.

The image generator continues on in this manner, and the switching lines60, 62, 64, 66, and 68 periodically repeat. However, the frequency ofswitching at any one line is low enough to reduce or eliminate theperception of false ghost objects.

Although this technique may generate false ghost objects at a switchingline, it has been empirically determined that because the switching lineeffectively moves from scanned image 50 to scanned image 50 such thatthe frequency of switching at a particular line is relatively low, thefalse ghost objects are less noticeable or unnoticeable. The occurrenceof false ghost objects may be further reduced if the period at which aparticular switch line repeats is greater than the persistence of thehuman eye. For example, if the time between a source-image switch at theline 66 and the following switch at the line 66 is greater than thepersistence of the human eye, a viewer is less likely to perceive falseghost objects near the line 66.

Furthermore, although an example where fs=8f_(v)/5 is discussed, thereare other relationships between f_(s) and f_(v) that reduce/eliminatefalse ghost objects. The optimum relationship between f_(s) and f_(v)and other artifacts often depends on the application and the actualvalues of f_(s) and f_(v), and thus is often determined on an ad hocbasis.

Modulating the Intensity of the Image Beam Relative to the Location ofthe Beam

Referring to FIG. 11, unless corrected, sinusoidal sweeping of an imagebeam may cause portions of a scanned image to appear brighter than otherportions.

FIG. 11 is a Lissajous pattern 70 that may cause the resulting scannedimage to have a non-uniform brightness. As discussed above inconjunction with FIGS. 4-8, an image generator (FIG. 17) scans thepattern 70 by sinusoidally sweeping an image beam bi-directionally inboth the horizontal (X) and vertical (Y) dimensions. Due to thesinusoidal sweep functions, the lines of the pattern 70 are closertogether, and thus more dense, toward the top region 72, bottom region74, and side regions 76 and 78 than they are in the center region 80.More specifically, because the top and bottom regions 72 and 74correspond to the peaks of the vertical sinusoid sweep function Y(t)(see equation (2) and FIGS. 5A and 5B), the beam travels more slowly inthe vertical (Y) dimension in these regions than it does in the centerregion 80. Therefore, because the beam strikes each unit of area in thetop and bottom regions 72 and 74 longer than it strikes each comparableunit of area in the center region 80, the image generator sweeps morelines, and thus forms more pixels of the scanned image, per unit area inthe top and bottom regions. Consequently, because the pixels are denserin the top 72 and bottom 74 regions, these regions appear brighter thanthe center region 80 if the image beam has a uniform maximum intensityover the entire pattern 70. Similarly, because the left 76 and right 78regions correspond to the peaks of the horizontal sinusoid sweepfunction X(t) (see equation (1) and FIGS. 5A and 5B), the beam travelsmore slowly in the horizontal (X) dimension in these regions. Therefore,because the pixels are denser in the left 76 and right 78 regions, theseregions appear brighter than the center region 80 if the beam has auniform intensity over the entire pattern 70.

A conventional technique for making the brightness of a scanned imageuniform where the beam is sinusoidally swept in the horizontal (X)dimension is to modulate the intensity of the beam in proportion to theinstantaneous sweep velocity in the horizontal (X) dimension. Therefore,in the side regions of the scanned image where the beam velocity islower, the beam intensity is proportionally lower, and in the centerregions where the beam velocity is higher, the beam intensity isproportionally higher. More specifically, because the horizontal sweepfunction X(t)=sin(2πf_(h)t+φ_(h)) denotes the horizontal location of thebeam, and using I max to denote the maximum instantaneous intensity ofthe beam, the modulated instantaneous intensity of the beam equalsI_(max) (instantaneous horizontal velocity)/(maximum horizontalvelocity)=I max×(d/dt sin(2πf_(h)t+φ_(h)))/max(d/dtsin(2πf_(h)t+φ_(h))), and is thus given by the following equation:I(modulated-maximum instantaneous beam intensity)=I _(max)×cos(2πf _(h)t+φ _(h))/1  (13)This horizontal modulation technique is further discussed in U.S. Pat.No. 6,445,362 to Tegreene, entitled “SCANNED BEAM DISPLAY WITH VARIATIONCOMPENSATION”, which is incorporated by reference. Intuitively, however,it may seem that modulating the beam intensity according the verticalsweep velocity would not provide the desired results where the verticalsweep frequency f_(v) is significantly lower than the horizontal sweepfrequency f_(h).

Still referring to FIG. 11, in one embodiment of the invention the imagegenerator (FIG. 17) makes the brightness of a scanned image more uniformby modulating the intensity of the beam in proportion to theinstantaneous sweep velocities in both the horizontal (X) and vertical(Y) dimensions. The inventors have determined that modulating the beamintensity according to the vertical sweep velocity does yield desirableresults. Therefore, in the top, bottom, and side regions 72, 74, 76, and78 of the scanned image where the beam velocity is lower, the beamintensity is proportionally lower, and in the center region 80 where thebeam velocity is higher, the beam intensity is proportionally higher.Specifically, because the vertical sweep functionY(t)=sin(2πf_(v)t+φ_(v)) denotes the vertical location of the beam, andusing I_(max) to denote the maximum instantaneous intensity of the beam,the modulated maximum instantaneous intensity I of the beam is given bythe following equation:I=I _(max)×cos(2πf _(h) t+φ _(h))×cos(2πf _(v) t+φ _(v))  (14)An alternative embodiment is to modulate the intensity of the beam inproportion to the vertical sweep velocity only such that:I=I _(max)×cos(2πf _(v) t+φ _(v))  (15)Conventional circuitry that can derive cos(2πf_(v)t+φ_(v)) and/orcos(2πf_(h)t+φ_(h)) from the sinusoidal sweeping functions X(t) and Y(t)of equations (1) and (2) and accordingly modulate the intensity of thebeam is relatively simple, thus making this modulation techniquerelatively easy to implement.

Referring to FIGS. 11 and 12, in another embodiment of the invention theimage generator (FIG. 17) sweeps the image beam more linearly in thevertical (Y) dimension to improve the brightness uniformity of a scannedimage.

FIG. 12 is a plot of the horizontal and vertical sweeping functions X(t)and Y(t) versus time for n_(v)=2, n_(h)=9, p_(v)=6, and p_(h)=8,although this embodiment of the invention can be used with other valuesof n_(v), n_(h), p_(v), and p_(h). Although the horizontal sweepingfunction X(t) is sinusoidal as discussed above in conjunction withequation (1) and FIGS. 4-8, the vertical sweeping function Y(t) is apseudo triangle wave with rounded peaks. By making the slopes of thevertical sweep function Y(t) more linear, the image generator (FIG. 17)sweeps the beam at a more constant velocity in the vertical (Y)dimension, thus making the line density, and thus the brightness, moreuniform in the top, bottom, and center regions 72, 74, and 80 of thepattern 70 (FIG. 11). More specifically, this embodiment of the verticalsweep function Y(t) is given by the following equation:Y(t)=(1−u)(p _(v)/2)sin(2πf _(v) t+φ _(v))+u(p _(v)/2)sin(2π3f _(v) t+φ_(v))  (16)where u is an empirically determined scale factor. One can see fromequation (16) that the more-linear slopes of the vertical sweep functionY(t) are obtained by adding a third harmonic of f_(v) to the sinusoidalY(t) of equation (2). And adding additional odd harmonics beyond thethird harmonic makes the slopes more linear by causing Y(t) to approacha triangle wave. Furthermore, as discussed below, one can design animage generator that vertically sweeps a beam according to equation (16)and compute the preferred phase relationship between X(t) of equation(1) and Y(t) of equation (16) according to the concepts discussed abovein conjunction with FIGS. 4-8.

Referring to FIGS. 11 and 12, other embodiments for improving thebrightness uniformity of the scanned image are contemplated. Forexample, one can vertically sweep the beam according to equation (15)and modulate the intensity of the beam in proportion to the horizontalsweep velocity, the vertical sweep velocity, or both the horizontal andvertical sweep velocities. Furthermore, one can modulate the beamintensity in proportion to a direct function of the beam location or asa function of a scanning angle of the beam instead of in proportion to asweep velocity which is the (derivative of the position and scanningangle) of the beam.

Referring again to FIG. 12, another advantage of making the line densityof a scan pattern such as the pattern 70 (FIG. 11) more uniform is thatit decreases the maximum line width Δ, and thus allows one to achieve adesired value for Δ with a lower horizontal sweep frequency f_(h). Thefollowing is a more general form of equation (7):Δ=(max vertical beam velocity)/2f _(h) n _(v)  (17)

where f_(v)<f_(n). Therefore, by reducing the maximum vertical beamvelocity, which is proportional to the maximum slope (i.e., the maximumof the time derivative) of the vertical sweep function Y(t), one canproportionally reduce f_(h) yet maintain Δ at a desired value. Becausethe maximum slope of the pseudo triangle wave of FIG. 12 is less thanthe maximum slope of a sinusoid (FIGS. 5A and 5B), using a pseudotriangle wave for Y(t) allows one to reduce f_(h) without increasing Δ.

Referring yet again to FIG. 12 and to equation (16), the phase andfrequency relationships discussed above in conjunction with FIGS. 4-8are determined according to an embodiment of the invention for avertical sweep function Y(t) that includes one or more harmonics off_(v). Specifically, equations (5), (6), (9), and (12) hold true for thefundamental frequency f_(v) of any such function Y(t). Furthermore,equation (12) can be modified for each harmonic of f_(v) by merelymultiplying the resulting possible phases by harmonic f_(v)/f_(v). Forexample, the third-harmonic phase equations corresponding to equations(11) and (12) are (equation (11) does not change):2πf _(h) t+φ _(h0)(the total phase of X(t))=±π/2  (11)2π3f _(v) t+φ _(v0)(the total phase of the third harmonic ofY(t))=3(π/2+(π/n _(h))[k+½]) for k=0,1, . . . (2n _(h)−1)  (18)

Consequently, one can design an image generator that uses amulti-harmonic vertical sweep function Y(t) according to the proceduresdiscussed above in conjunction with FIGS. 4-8. Moreover, using the sameprinciples, one can design an image generator that uses a multi-harmonichorizontal sweep function X(t).

Interpolating the Intensities of the Scanned Pixels from the SourcePixels

Referring to FIGS. 13-15, because a sinusoidal scanning patterntypically does not intersect the locations of the source pixels from thesource image, the locations of the scanned pixels typically do notcoincide with the locations of the source pixels. Consequently, theimage generator (FIG. 17) may interpolate the intensities of the scannedpixels from the intensities of the source pixels to improve the qualityof the scanned image.

FIG. 13 is a plot of the bi-sinusoidal scanning pattern 40 and the gridpattern 42 of FIG. 4, and illustrates a technique for forming thescanned pixels Z on the vertical grid lines and interpolating theirintensities from the vertically adjacent source pixels P according to anembodiment of the invention.

To locate the scanned pixels Z coincident with the vertical lines of thegrid pattern 42, the image generator (FIG. 17) generates a non-linearpixel clock that indicates when the image beam intersects a verticalgrid line. Because the horizontal sweep function X(t) is non-linear—herea sinusoid per equation (5)—the time from when the beam intersects avertical grid line to the time when it intersects an immediatelyadjacent vertical grid line differs from grid line to grid line. Forexample, it takes longer for the beam to travel between the grid lines 3and 4 (pixels Z_(4,y) and Z_(3,y)) than it does for the beam to travelbetween the grid lines −1 and 1 (pixels Z_(1,y) and Z_(1,y)). This isbecause the beam travels more slowly in the horizontal (X) dimensionnear the sides of the pattern 40—the sides correspond to the peaks ofthe horizontal sinusoid—than it does near the center—the centercorresponds to the zero crossings of the horizontal sinusoid. Therefore,the image generator generates the pixel clock such that itsinstantaneous period is proportional to the horizontal velocity of thebeam. As a result, the pixel clock “ticks” whenever the beam intersectsa vertical grid line (or a predetermined offset time before thisintersection) regardless of the horizontal position of the beam. Atechnique for generating such a pixel clock is disclosed in previouslyincorporated U.S. Pat. No. 6,140,979.

Because the scanned pixels Z are coincident with the vertical lines ofthe grid pattern 42, the image generator interpolates the intensity ofeach pixel Z from the source pixels P that are immediately above andbelow the pixel Z on the same vertical grid line. For example, the imagegenerator interpolates the intensity of the pixel Z_(1,y) from theintensities of the source pixels P_(1,1) and P_(1,−1). In oneembodiment, the image generator computes the intensity IZ_(1,y) ofZ_(1,y) according to the following conventional linear-interpolationequation:IZ _(1,y) =αIP _(1,1)+(1−α)IP _(1,−1)  (19)where α is the absolute value of the vertical distance between P_(1,−1)and Z_(1,y), (1−α) is the absolute value of the vertical distancebetween P_(1,1) and Z_(1,y), and IP_(1,−1) and IP_(1,−1) are therespective intensities of the source periods P_(1,−1) and P_(1,1). Theimage generator typically retrieves the intensities of the pixels P fromthe buffer (FIG. 17) that stores the corresponding source image.Alternatively, the image generator may use other conventionalinterpolation algorithms.

To interpolate the intensities of the scanned pixels Z from the adjacentsource pixels P as discussed in the preceding paragraph, the imagegenerator tracks the position of the image beam relative to the grid 42so that it can determine which pixels P to use for the interpolation.Techniques for tracking the position of the image beam are discussedbelow.

Still referring to FIG. 13, one technique for tracking the horizontalposition of the image beam is to clock a horizontal-position counterwith the non-linear pixel clock. For example, when the beam starts atthe left edge of the pattern 40, the clock can store an initial count ofzero. Then, as the beam moves toward the right and intersects thevertical grid line p_(h)=−4, the pixel clock “ticks” to increment thecount by one, and to thus indicate the first pixel in the horizontaldimension. This incrementing continues for each vertical grid line untilthe pixel clock increments the count to eight when the beam intersectsthe vertical grid line p_(h)=4. Next, as the beam intersects thevertical grid line p_(h)=3 on its way back from the right edge of thepattern 40, the pixel clock “ticks” to decrement the count by one, andto thus indicate the seventh pixel in the horizontal direction. Thisdecrementing continues for each vertical grid line until the pixel clockdecrements the count back to zero. Then this increment/decrement cyclerepeats for each subsequent cycle of the horizontal sweep function X(t).

The image generator can track the vertical position of the image beam ina similar manner by generating a non-linear vertical pixel clock andclocking a vertical position counter therewith. To provide a measure ofα, the frequency of the non-linear vertical pixel clock can be increasedby a scale factor. For example, increasing the frequency by ten providesten clock “ticks” between each pair of pixels P in the verticaldimension, and thus provides a to a resolution of 0.1 pixels.

Another technique for keeping track of the horizontal and verticalpositions of the image beam is discussed below in conjunction with FIG.15. Other techniques are also available, but are omitted for brevity.

Referring to FIG. 14, because generating a non-linear pixel clock oftenrequires a relatively large and complex circuit, the image generator(FIG. 17) may use a linear pixel clock (a clock having a constantperiod) to interpolate the intensities of the scanned pixels Z asdiscussed below.

FIG. 14 is a plot of a section 90 of the grid pattern 42 of FIG. 13 anda scanned pixel Z having an arbitrary location within this grid sectionaccording to an embodiment of the invention. Because the linear pixelclock does not force the scanned pixels Z to coincide with the verticallines of the grid pattern 42, the pixel Z can have any arbitrarylocation x+β (horizontal component), y+α (vertical component) within thegrid section. Therefore, the image generator (FIG. 17) interpolates theintensity IZ of Z_(x+β,y+α) from the intensities of the surrounding foursource pixels P according to the following conventional bi-linearinterpolation equation:IZ=(1−α)[(1−β)IP _(x,y) +βIP _(x+1,y)]+α[(1−β)IP _(x,y+1) +βIP_(x+1,y+1)]  (20)Equation (20) is valid regardless of the direction in which the scanassembly (FIG. 17) is sweeping the image beam as it forms the pixelZ_(x+β,y+x). Alternatively, the image generator may interpolate theintensity IZ according to another interpolation algorithm that usesthese four source pixels P, a subset of these four source pixels, othersource pixels, or a combination of these source pixels and other sourcepixels.

FIG. 15 is a block diagram of a position-tracking and interpolationcircuit 100 that is operable to track the horizontal and verticalpositions x+β and y+α of a bi-sinusoidally-swept image beam according toan embodiment of the invention. The circuit 100 includes a pixel clockcircuit 102, horizontal and vertical phase accumulators 104 and 106,horizontal and vertical position accumulators 108 and 110, a memory 112,horizontal and vertical position translators 114 and 116, and aninterpolator 118. As discussed further below, the pixel-clock circuit102 generates a linear pixel clock having a constant clock period. Thephase accumulators 104 and 106 respectively track the phases of thehorizontal and vertical sweep functions X(t) and Y(t) (equations (1) and(2)). The position accumulators 108 and 110 respectively calculate thehorizontal and vertical positions of the image beam from the horizontaland vertical phases and from sweep-function-trajectory approximationsfrom the memory 112. The translators 114 and 116 respectively translatethe horizontal and vertical positions into the coordinates of thesource-image grid pattern such as the pattern 42 of FIG. 13, and theinterpolator 118 calculates the intensity of a scanned pixel Z_(x+β,y+α)from the translated horizontal and vertical positions and respectivesource pixels P.

The clock circuit 102 generates a linear pixel clock having a frequencyf_(p) according to the following equation:f_(p)=Mf_(h)  (21)where M=2p_(h). For example, referring to FIG. 13, assume that the phaseof the horizontal sweep function X(t) equals −π/2 at the left side ofthe scanning pattern 40 and +π/2 at the right side, and that p_(h)=8pixels. As the image generator (FIG. 17) sweeps the image beam through afull horizontal cycle of 2π radians (left side to right side and back tothe left side) it generates eight pixels Z during the left-to-rightsweep and another eight pixels Z during the right-to-left sweep. Asdiscussed above, because each “tick” of the pixel clock identifies theinstant for generating a respective pixel Z, the pixel clock includessixteen “ticks” per horizontal cycle, and thus equals 16f_(h) (sixteentimes the horizontal sweep frequency) in this example.

The horizontal and vertical phase accumulators 104 and 106 respectivelytrack the total phases θ and ψ of the horizontal and vertical sweepfunctions X(t) and Y(t) according to the following equations:θ_(n)=θ_(n-1)+2π/M  (22)ψ_(n)=ψ_(n-1)+(n _(v) /n _(h))2π/M  (23)where n represents the current “tick” of the pixel clock and n−1represents the immediately previous “tick”. For example, where thehorizontal and vertical sweep functions X(t) and Y(t) are sinusoids perequations (1) and (2)θ=2πf _(h)+φ_(h)  (24)ψ=2πf _(v)+φ_(v).  (25)Because the phase of a sinusoid increases linearly versus time, for each“tick” of the pixel clock the horizontal phase θ increases by the sameamount 2π/M. For example, where M=16, the horizontal phase θ incrementsπ/8 radians for each “tick”, and thus completes a full rotation of 2πevery sixteen “ticks”, which is equivalent to one cycle of thehorizontal sweep frequency f_(h) per above. Furthermore, becausef_(v)=f_(h)n_(v)/n_(h) (equation (6)), the vertical phase ψ onlyincreases n_(v)/n_(h) as fast as θ does. For example, where n_(v)=2,n_(h)=9, and M=16, ψ increments ( 2/9)×π/8=π/36 radians for each “tick”,and thus repeats a full rotation of 2π every seventy two “ticks” of thepixel clock, which is every four and one half cycles of the horizontalsweep frequency f_(h). This agrees with equation (6) and FIGS. 5A and5B. Moreover, in one embodiment, θ_(n) and ψ_(n) overflow to zero whenthey reach 2π.

Referring to FIGS. 15 and 16, the horizontal and vertical positionaccumulators 108 and 110 respectively track the horizontal and verticalpositions X and Y of the image beam according to the followingequations:X _(n) =X _(n-1) +a _(j)2π/M  (26)Y _(n) =Y _(n-1) +c _(i)(n _(v) /n _(h))2π/M  (27)where a_(j) represents a linear approximation of the horizontal sweepingfunction X(t) at 2π/M, and c_(i) represents a linear approximation ofthe vertical sweeping function Y(t) at (n_(v)/n_(h))2π/M. For example,where X(t) and Y(t) are sinusoidal per equations (1) and (2) above, theycan be conventionally represented by respective Taylor series expansionsthat effectively break down the sinusoids into a number (_(j, i)) ofline segments. The more line segments used, the more accurate the linearapproximation. A_(j) and c_(i) are the slopes of these line segments inunits of distance (in terms of source pixels) per radian, where j=0 tothe j_(th) line segment approximating X(t), and i=0 to the i_(th) linesegment approximating Y(t). Therefore, a_(j)2π/M is the horizontaldistance in pixels p_(h) traveled by the beam in one clock “tick”, andc_(i)(n_(v)/n_(h))2π/M is the vertical distance in pixels p_(v) traveledby the beam in one clock “tick.” The horizontal and vertical positionaccumulators 108 and 110 respectively retrieve a_(j) and c_(i) from thememory 112 based on the horizontal and vertical phases θ_(n) and ψ_(n),and store the retrieved a_(j) and c_(i) until θ_(n) and ψ_(n) cause theaccumulators 108 and 110 to update a_(j) and c_(i) by retrieving updatedvalues therefore.

Still referring to FIGS. 15 and 16, an example is presented toillustrate the operation of the horizontal positional accumulator 108according to an embodiment of the invention. FIG. 16 is a plot of thehorizontal and vertical sweep sinusoids X(t) and Y(t) of FIG. 5A andtheir respective linear approximations where j=i=0,1. That is, X(t) isapproximated with two line segments j=0 and j=1 respectively havingslopes a₀=(8 pixels)/(π radians) and a₁=−(8 pixels)/(π radians).Therefore, the horizontal position accumulator 108 uses a₀ in equation(26) for −π/2<θ_(n)≦+π/2, and uses a₁ in equation (26) for +π/2<θn≦−π/2.Specifically, when θ_(n) transitions from being less than or equal to−π/2 to being greater than −π/2, the horizontal position accumulator 108retrieves a₀ from the memory 112, and stores a₀ for repeated use untilθ_(n) becomes greater than +π/2. And when On becomes greater than +π/2,the accumulator 108 retrieves a₁ from the memory 112, and stores a₁ forrepeated use until θ_(n) again becomes greater than −π/2. Consequently,the accumulator 108 need only access the memory 112 twice per horizontalsweep cycle.

Using the above example, the vertical position accumulator 110 operatesin a similar manner. Y(t) is approximated by two line segments i=0 andi=1 respectively having slopes c₀=(6 pixels)/(π radians) and c₁=−(6pixels)/(π radians). The accumulator 110 uses c₀ in equation (27) for−π/2<ψ_(n)≦+π/2 and uses c₁ in equation (27) for +π/2<ψ_(n)≦−π/2.Specifically, when ψ_(n) transitions from being less than or equal to−π/2 to being greater than −π/2, the vertical position accumulator 110retrieves c₀ from the memory 112, and stores c₀ for repeated use untilψ_(n) becomes greater than +π/2. And when ψ_(n) becomes greater than+π/2, the accumulator 110 retrieves c₁ from the memory 112, and storesc₁ for repeated use until ψ_(n) again becomes greater than −π/2.Consequently, the accumulator 110 need only access the memory 112 twiceper vertical sweep cycle.

Referring again to FIG. 15, the horizontal and vertical positiontranslators 114 and 116 respectively shift the horizontal and verticalpositions X and Y to be compatible with the grid pattern 42 (FIG. 13)according to the following equations:X _(translated) =X _(n) +P _(h)/2−0.5+L _(h)  (28)Y _(translated) =Y _(n) +P _(v)/2−0.5+L _(v)  (29)where L_(h) and L_(v) are optional alignment-correction factors asdiscussed below.

Specifically, X_(n) and Y_(n) are incompatible with the grid pattern 42(FIG. 13) because they are in terms of the amplitudes of the sweepfunctions X(t) and Y(t) in units of pixels. For example, where p_(h)=8and P_(v)=6 as in FIG. 13 and X(t) and Y(t) are sinusoidal, X_(n) rangesfrom −4 to +4 pixels and Y_(n) ranges from −3 to +3 pixels per theamplitudes of the horizontal and vertical sweep sinusoids (FIG. 16).

To be compatible with the grid pattern 42 (FIG. 13), it is desirablethat X_(translated) range from −0.5 to +7.5, and that Y_(translated)range from −0.5 to 5.5. Therefore, equations (28) and (29) respectivelyobtain these preferred ranges for X_(translated) and Y_(translated) byeffectively shifting X_(n) by P_(h)/2−0.5=3.5 pixels and effectivelyshifting Y_(n) by P_(v)/2−0.5=2.5 pixels.

L_(h) and L_(v) of equations (28) and (29), respectively, mathematicallyaccount for misalignment of the image beam. In one embodiment of theinvention, the horizontal and vertical phase accumulators 104 and 106are respectively calibrated such that θ_(n)=0 as the reflector (FIG.17), rotates through its horizontal 0π position, and such that ψ_(n)=0as the reflector rotates through its vertical 0π position—the reflectorposition is measured using conventional techniques such as discussed inU.S. Pat. No. 5,648,618 to Neukermans entitled “MICROMACHINED HINGEHAVING AN INTEGRAL TORSIONAL SENSOR”, which is incorporated byreference. But if the image beam does not strike the respectivehorizontal or vertical center of the display screen (FIG. 1) as thereflector rotates through its horizontal and vertical 0π positions, thenthe actual position of the beam is offset from the position indicated bythe reflector position. Because this offset can typically be measuredand is often substantially constant regardless of the beam position, itsx and y components are respectively represented by the constants L_(h)and L_(v), which have units of pixels. That is, including L_(h) andL_(v) in equations (28) and (29) insure that X_(translated) andY_(translated) represent the actual position of the beam, not merely theposition of the reflector. This technique is particularly useful wherethe image generator (FIG. 17) sweeps three misaligned beams, such as red(R), green (G), and blue (B) beams, to scan a color image. Bycalculating separate X_(translate) and Y_(translate) values for eachbeam, the image generator can mathematically correct for thismisalignment during interpolation of the scanned pixels Z (FIG. 13) byusing appropriate values for L_(hred), L_(vred), L_(hgreen), L_(vgreen),L_(hblue), and L_(vblue).

In one embodiment of the invention the horizontal and vertical positiontranslators 114 and 116 are floating-point counters such that theinteger portions of X_(translated) and Y_(translated) are thecoordinates of the lowest-number source pixel P and the decimal portionsare β and α, respectively. For example, referring to FIG. 14, where theimage beam is positioned to form the scanned pixel Z_(x+β,y+α), theinteger portions of X_(translated) and Y_(translated) equal x and y,respectively, and the decimal portions equal β and α, respectively.

Still referring to FIG. 15, the interpolator 118 interpolates theintensities of the scanned pixels Z from the values ofX_(translated)=x+β and Y_(translated)=y+α in a conventional manner suchas that discussed above in conjunction with FIG. 14.

Other embodiments of the position-tracking and interpolation circuit 100are contemplated. For example, because the scanning pattern 40 (FIG. 13)repeats itself and because the period of the pixel clock, and thus thelocations of the pixels Z, are known in advance, all possible values ofX_(translated) and Y_(translated) can be determined in advance andstored in a look-up table (not shown). Therefore, the interpolator 118need only retrieve X_(translated) and Y_(translated) from this look-uptable. Such a circuit, however, would include two memory accesses perpixel clock period. This is unlike the circuit 100 of FIG. 15, where thehorizontal and vertical position accumulators 108 and 110 access thememory 12 only when changing from one line segment j and i, and thusfrom one slope a_(j) and c_(i), to another. In addition, the circuit 100may calculate only the vertical position y+α if a non-linear horizontalpixel clock is used per FIG. 13 such that the pixels Z are aligned withthe vertical lines of the grid 42, i.e., β is always zero. In such anembodiment, the horizontal translator 114 can be replaced with a counterthat, along with the interpolator 118, is clocked with the non-linearhorizontal pixel clock, while the circuits 106, 110, and 116 are clockedwith the linear pixel clock generated by the circuit 102.

Image Generator

FIG. 17 is a block diagram of an image generator 130 that can implementthe above-described techniques according to an embodiment of theinvention. The image generator 130 includes a scan assembly 132, animage-beam generator 134 for generating an image beam 136, and asource-image buffer 138.

The scan assembly 132 includes a sweep-drive circuit 140 and aconventional reflector 142, such as the reflector 22 of FIG. 1. Thecircuit 140 can drive the reflector 142 such that the reflector sweepsthe beam 136 bi-sinusoidally and/or bi-directionally in the verticaldimension as discussed above in conjunction with FIGS. 4, 5A, 5B, and12.

The image-beam generator 134 includes a position-intensity circuit 144,a scanned-pixel interpolator 146, a conventional beam source 148, and abuffer-switch circuit 150. The circuit 144 can modulate the intensity ofthe beam 136 according to the position of the beam as discussed above inconjunction with FIGS. 11 and 12. The interpolator 146 can modulate theintensity of the beam 136 to interpolate the intensities of the scannedpixels Z as discussed above in conjunction with FIGS. 13-16, and mayinclude the circuit 100 of FIG. 15. The beam source 148 generates thebeam 136, and may be, e.g., a light-emitting diode (LED) or a laserdiode. The switch circuit 150 transitions the generation of the imagebeam 136, and thus the formation of the scanned pixels 2, from onesource image in the buffer 138 to another source image in the buffer toreduce or eliminate the perception of false ghost images as discussedabove in conjunction with FIGS. 9 and 10.

The source-image buffer 138 is a conventional buffer that receivessource video or still images in a conventional manner from aconventional source. For example, the buffer 138 may receive videoimages via a stream of video data from a computer (not shown) or fromthe internet (not shown).

Still referring to FIG. 17, other embodiments of the image generator 130are contemplated. For example, the beam 136 may be an electron beam fordisplay on a phosphor screen of a cathode-ray tube (CRT), and thereflector may be a coil or other device for sweeping the beam. And wherethe beam 136 is a light beam, the reflector may direct it onto a displayscreen (FIG. 1) or directly into a viewer's eye (not shown).Furthermore, although “horizontal” and “vertical” have been used aboveto denote orthogonal side-to-side and up-down dimensions, respectively,they may denote other respective dimensions that may not be orthogonal.For example, “vertical” may generally denote the dimension having thelower sweep frequency, even if this is not the up-down dimension.

The preceding discussion is presented to enable a person skilled in theart to make and use the invention. Various modifications to theembodiments will be readily apparent to those skilled in the art, andthe generic principles herein may be applied to other embodiments andapplications without departing from the spirit and scope of the presentinvention. Thus, the present invention is not intended to be limited tothe embodiments shown, but is to be accorded the widest scope consistentwith the principles and features disclosed herein.

1. A method, comprising: Sweeping an image beam in a horizontaldimension according to a horizontal sinusoid having a phase; andsweeping the image beam back and fourth in a vertical dimensionaccording to a vertical sinusoid, wherein the image beam is swept whilethe image beam is “on” from a first edge of an image to a second edge ofthe image continuously in a first direction of the vertical dimension,and wherein the image beam is swept while the image beam is “on,” fromthe second edge of the image to the first edge of the image continuouslyin a second direction of the vertical dimension, such that the imagebeam traverses a pattern that repeats every n_(h) periods of thehorizontal sinusoid, the vertical having a phase equal to${- \frac{\pi}{2}} + {\frac{\pi}{n_{h}}\left\lbrack {k + \frac{1}{2}} \right\rbrack}$when the phase of the horizontal sinusoid equals $\pm \frac{\pi}{2}$ andwhere k=0, 1, . . . (2n_(h)−1).
 2. The method of claim 1 whereinsweeping the image beam in the horizontal dimension comprises sweepingthe image beam back and forth in the horizontal dimension.
 3. The methodof claim 1 wherein: sweeping the image beam in the horizontal dimensioncomprises sinusoidally sweeping the image beam in the horizontaldimension at a horizontal frequency; and sweeping the image beam in thevertical dimension comprises sinusoidally sweeping the image beam backand forth in the vertical dimension at a vertical frequency that islower than the horizontal frequency.
 4. The method of claim 1 whereinsweeping the image beam back and forth in the vertical dimensioncomprises sweeping the image beam along a path that is a function of afundamental vertical frequency and a harmonic of the fundamentalvertical frequency.
 5. The method of claim 1 wherein the horizontaldimension is substantially orthogonal to the vertical dimension.
 6. Themethod of claim 1, wherein sweeping the image beam in the horizontaldimension comprises sweeping the image beam bi-directionally in thehorizontal dimension.
 7. The method of claim 1, further comprising: afirst reflector sweeping the image beam in the horizontal dimension; anda second reflector sweeping the image beam in the vertical dimension. 8.The method of claim 1, further comprising a single reflectorsimultaneously sweeping the image beam in the horizontal dimension andthe vertical dimension.
 9. The method of claim 1 further comprising:sweeping the image beam in the horizontal dimension by resonating at ahorizontal frequency; and sweeping the image beam in the verticaldimension by resonating at a vertical frequency.
 10. The method of claim1 having horizontal and vertical resonant frequencies and furthercomprising sweeping the image beam in the horizontal dimension byresonating at the horizontal frequency; and sweeping the image beam inthe vertical dimension by oscillating at a vertical frequency other thanthe vertical resonant frequency.
 11. A method, comprising: Sweeping animage beam back and forth in the vertical dimension according to avertical sinusoid having a phase; and sweeping the image beam back andfourth in a horizontal dimension according to a horizontal sinusoid,wherein the image beam is swept while the image beam is “on” from afirst edge of an image to a second edge of the image continuously in afirst direction of the horizontal dimension, and wherein the image beamis sweet while the image beam is “on,” from the second edge of the imageto the first edge of the image continuously in a second direction of thehorizontal dimension, such that the image beam traverses a pattern thatrepeats every n_(v) periods of the horizontal sinusoid, the verticalhaving a phase equal to${- \frac{\pi}{2}} + {\frac{\pi}{n_{v}}\left\lbrack {k + \frac{1}{2}} \right\rbrack}$when the phase of the horizontal sinusoid equals ±π/2 and where k=0, 1,. . . (2n_(v)−1).
 12. The method of claim 11 wherein: sweeping the imagebeam in the vertical dimension comprises sinusoidally sweeping the imagebeam in the vertical dimension at a vertical frequency; and sweeping theimage beam in the horizontal dimension comprises sinusoidally sweepingthe image beam back and forth in the horizontal dimension at ahorizontal frequency that is lower than the vertical frequency.
 13. Themethod of claim 11 wherein sweeping the image beam back and forth in thehorizontal dimension comprises sweeping the image beam along a path thatis a function of a fundamental horizontal frequency and a harmonic ofthe fundamental horizontal frequency.
 14. The method of claim 11 whereinthe horizontal dimension is substantially orthogonal to the verticaldimension.
 15. The method of claim 11, wherein sweeping the image beamin the vertical dimension comprises sweeping the image beambi-directionally in the vertical dimension.
 16. The method of claim 11,further comprising: a first reflector sweeping the image beam in thehorizontal dimension; and a second reflector sweeping the image beam inthe vertical dimension.
 17. The method of claim 11, further comprising asingle reflector simultaneously sweeping the image beam in thehorizontal dimension and the vertical dimension.
 18. The method of claim11 further comprising: sweeping the image beam in the horizontaldimension by resonating at a horizontal frequency; and sweeping theimage beam in the vertical dimension by resonating at a verticalfrequency.
 19. The method of claim 11 having horizontal and verticalresonant frequencies and further comprising sweeping the image beam inthe vertical dimension by resonating at the vertical frequency; andsweeping the image beam in the horizontal dimension by oscillating at ahorizontal frequency other than the horizontal resonant frequency. 20.An image generator comprising: a scan assembly operable to: Sweep animage beam in a horizontal dimension according to a horizontal sinusoidhaving a phase; and sweeping the image beam back and fourth in avertical dimension according to a vertical sinusoid, wherein the imagebeam is swept while the image beam is “on” from a first edge of an imageto a second edge of the image continuously in a first direction of thevertical dimension, and wherein the image beam is swept while the imagebeam is “on,” from the second edge of the image to the first edge of theimage continuously in a second direction of the vertical dimension, suchthat the image beam traverses a pattern that repeats every n_(h) periodsof the horizontal sinusoid, the vertical having a phase equal to${- \frac{\pi}{2}} + {\frac{\pi}{n_{h}}\left\lbrack {k + \frac{1}{2}} \right\rbrack}$when the phase of the horizontal sinusoid equals $\pm \frac{\pi}{2}$ andwhere k=0, 1, . . . (2n_(h)−1).
 21. An image generator comprising: ascan assembly operable to: Sweep an image beam back and forth in thevertical dimension according to a vertical sinusoid having a phase; andsweeping the image beam back and fourth in a horizontal dimensionaccording to a horizontal sinusoid, wherein the image beam is sweptwhile the image beam is “on” from a first edge of an image to a secondedge of the image continuously in a first direction of the horizontaldimension, and wherein the image beam is sweet while the image beam is“on,” from the second edge of the image to the first edge of the imagecontinuously in a second direction of the horizontal dimension, suchthat the image beam traverses a pattern that repeats every n_(v) periodsof the horizontal sinusoid, the vertical having a phase equal to${- \frac{\pi}{2}} + {\frac{\pi}{n_{v}}\left\lbrack {k + \frac{1}{2}} \right\rbrack}$when the phase of the horizontal sinusoid equals ±π/2 and where k=0, 1,. . . (2n_(v)−1).